Stanford Economist Ken Arrow
received the Nobel Prize in 1972 for proving (in 1951) that there is
no such thing as a perfect voting system. This means that for
any system one can come up with an example that makes that system
look bad. The question, therefore, is not whether one can come
up with an example, but rather how realistic that example is.
A second question is whether one has any real examples of absurd
outcomes, instead of invented ones.

This page lists examples that illustrate flaws with a few
commonly discussed voting systems:

We leave it up to you to judge how realistic these scenarios
are and whether you can find any real-world examples of actual
elections that involved such scenarios.

Plurality

Plurality voting has a flaw that occurs
frequently: when a small number of people vote for a third
party candidate, they can swing a close race to the major party
candidate opposed by a majority of voters. This happened in the 2000
presidential with Democratic candidate Al Gore and Green Party
candidate Ralph Nader both in the state of Florida and
nationally. It also occurred in the states of New Mexico and
Iowa, where Pat Buchanan supporters swung the states to Gore and in
a US Senate and a state senate race in Washington State,
where Libertarian candidates swung the race to Democrats, which
ended giving Democrats control of both the state senate and the US
senate, after Jim Jeffords defection from the Republican Party.

Approval voting

In approval voting, voters can vote for or approve of
as many candidates as they like, and the candidate with the most
votes wins the election.

Under a two-round or instant runoff, no one has a
majority after the first round, so White is eliminated, her
supporters prefer Smith, and Smith is elected 60% to 40%. This
makes sense. White had little enthusiastic support, and when the
race went to Jones versus Smith, most of the voters preferred
Smith.

But what happens under Approval Voting? If each
of the factions does not bullet vote, but rather approves of their
top two choices, Jones gets 75% of the vote, White gets 65% of the
vote, and Smith gets only 60% of the vote! Even though most
voters prefer Smith to Jones, Jones wins!

Well, when the uproar
dies down, what does Smith do in the next election? He asks
his voters to bullet vote, of course. After all, if his
enthusiastic supporters had simply bullet voted the first time, he
would have had a better chance of winning; his voters caused Jones
to win. (Obviously, if only Smith's voters bullet vote, and
everyone else votes the same way in the second election as in the
first, White will win. This will cause Jones to also tell her
supporters to bullet vote, either when she gets wind of Smith's
strategy, or in the third election.)

Thus we see that this sort of thing is
contagious, as anyone who has been around politics (as opposed to
academia) knows, and before long, everyone asks their supporters
to bullet vote, causing Approval Votingto
degenerate into simple plurality. If this didn't happen in the
first election, it would happen in the second and subsequent
elections.

So in this situation, you have to use another method
to eliminate one of the candidates to determine the winner.

In addition to the flaws revealed by specific
examples, approval and Condorcet voting suffer from another real
world problem: they encourage candidates to avoid taking
public stands on controversial issues, since the least offensive
candidate can emerge victorious. Thus, compared to instant
runoff voting, they create a less-well informed electorate.

Instant runoff voting

A commonly cited example is one in which two extreme candidates have
strong core support, neither can appeal to a majority, and
a compromise candidate has weak core support but is preferred
by a majority over the other two candidates. For example:

Candidate
SupportJones
45%Marvin
Moderate 15%Smith
40%

If the supporters of the extreme candidates prefer the moderate to the
other extremist, then the moderate candidate would win head-to-head races
against both of the other candidates. In an instant
runoff or a two-round runoff with this example, the compromise candidate
is eliminated, and one of the extremists is
elected.

As an astute visitor pointed out, instant runoff
voting generally does a better job of finding the true compromise
candidate than either plurality or two-round runoff elections.

Borda Count

In an election with n candidates using a Borda Count, a candidate receives
n-1 points for each 1st choice, n-2 for each 2nd choice and so
on. The candidate with the most points wins the
election.

The chief flaw in this system is that your vote for
your 2nd choice can end up defeating your first
choice.